Question

Solve the differential equation $\log\left(\dfrac{dy}{dx}\right) = 3x + 4y$.

Hint:

Always use separation of variables for equations of the form $\dfrac{dy}{dx} = f(x)\,g(y)$.

Answer 1

Step 1: Rewrite equation. \[ \log\left(\dfrac{dy}{dx}\right) = 3x + 4y \] Take exponential both sides: \[ \dfrac{dy}{dx} = e^{3x + 4y} \]

Step 2: Separate variables. \[ e^{-4y} dy = e^{3x} dx \]

Step 3: Integrate both sides. \[ \int e^{-4y} dy = \int e^{3x} dx \] \[ \dfrac{e^{-4y}}{-4} = \dfrac{e^{3x}}{3} + C \]

Step 4: Simplify. \[ e^{-4y} = -\dfrac{4}{3} e^{3x} + C' \] (where $C' = 4C$ is constant of integration).

Final Answer: \[ \boxed{e^{-4y} + \dfrac{4}{3}e^{3x} = C} \]